536 CHAPTER 5 Trigonometric Functions We can find the six trigonometric functions using any point other than the origin on the terminal side of an angle. To see why any point can be used, refer to Figure 16, which shows an angle u and two distinct points on its terminal side. Point P has coordinates 1x, y2, and point P′ (read “P-prime”) has coordinates 1x′, y′2. Let r be the length of the hypotenuse of triangle OPQ, and let r ′ be the length of the hypotenuse of triangle OP′Q′. Because corresponding sides of similar triangles are proportional, we have y r = y′ r′ . Corresponding sides are proportional. Thus sin u = y r is the same no matter which point is used to find it. A similar result holds for the other five trigonometric functions. We can also find the trigonometric function values of an angle if we know the equation of the line coinciding with the terminal ray. Recall that the graph of the equation Ax + By = 0 Linear equation in two variables x and y is a line that passes through the origin 10, 02. If we restrict x to have only nonpositive or only nonnegative values, we obtain as the graph a ray with endpoint at the origin. For example, the graph of x + 2y = 0, x Ú 0, shown in Figure 17, is a ray that can serve as the terminal side of an angle u in standard position. By choosing a point on the ray, we can find the trigonometric function values of the angle. x y 0 (2, –1) x + 2y = 0, x ≥ 0 x = 2 y = –1 r = √5 U Figure 18 x y 0 x + 2y = 0, x ≥ 0 U 2 Figure 17 x y (x9, y9) OP = r OP9 = r9 O Q Q9 P P9 (x, y) u Figure 16 EXAMPLE 3 Finding Function Values of an Angle Find the six trigonometric function values of an angle u in standard position, if the terminal side of u is defined by x + 2y = 0, x Ú 0. SOLUTION The angle is shown in Figure 18. We can use any point except 10, 02 on the terminal side of u to find the trigonometric function values. We choose x = 2 and find the corresponding y-value. x + 2y = 0, x Ú 0 2 + 2y = 0 Let x = 2. 2y = -2 Subtract 2. y = -1 Divide by 2. The point 12, -12 lies on the terminal side, and so r = 222 + 1-122 = 25. Now we use the definitions of the trigonometric functions. sin u = y r = -12 5 = -12 5 # 25 25 = - 25 5 cos u = x r = 22 5 = 22 5 # 25 25 = 225 5 tan u = y x = -1 2 = - 1 2 csc u = r y = 25 -1 = - 25 sec u = r x = 25 2 cot u = x y = 2 -1 = -2 S Now Try Exercise 43. Multiply by 25 25 , a form of 1, to rationalize the denominators.
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